Overview:Each player tries to connect their two edges of the board by drawing dots and connecting them together.

Each turn:* A player draws a dot in ANY empty square on the board.* Then the player connects it (draws a line) to all of their adjacent dots in the eight adjoining squares, unless it would cross an opponent's connection.

Win condition: A player connects their two edges of the board.

Pie rule (optional, this rule nullifies the 1st player's inherent advantage): After the first move is made, the second player has one of two options:

1) Letting the move stand, in which case the second player remains the second player and moves immediately, or2) Switching places, in which case the second player becomes the first-moving player, and the "new" second player then makes their "first" move. (I.e., the game proceeds from the opening move already made, but with roles reversed.)

Now the puzzle:

In the following 7x7 n00b game, Red moved at d3, then Blue at b4, then Red at d5. So it's Blue's turn to move. Is there any way:

1a) Blue can stop red from forming a connection between d3 and d5?1b) If not, where should have Blue moved instead of b4 on their last turn?2) Can Blue stop Red from winning the game at this point (i.e., has Red already won)?

Some questions:Do the lines connecting dots have to be straight, or are they free-form?How thick are the lines?Do they have to go between the centers of the dots, (i.e. perpendicular to the edge of the dot), or can they come out at any angle?

gmalivuk wrote:

King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.

Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

Some questions:Do the lines connecting dots have to be straight, or are they free-form?

Straight.

How thick are the lines?

As thick as the line your marker, pen, crayon, colored pencil, etc makes.

Do they have to go between the centers of the dots, (i.e. perpendicular to the edge of the dot), or can they come out at any angle?

So to explain it in a bit more detail:Each turn you draw a dot in an empty square. Then if there is a pre-existing dot of yours in any of 8 squares surrounding it, you draw a line between the dot you just made and that dot. You do this for any and all dots in the 8 squares around your 'new' dot - so you could possibly draw up to 8 straight lines radiating from your new dot on a turn. However, if there's an opponent's connection already in the way of drawing a connection, then you don't draw it.

I hope that made sense.

SirGabriel: Yes, you are correct I believe. There is also another thing the 2nd player (hint hint) could have done on their first turn that would have helped their chances greatly, especially on a small board

If there's interest, I'll post another problem for this game later tonight or tomorrow.

We can see that there exists a strategy by which the first player can always win by "strategy stealing" argument.

1) There must be a winner when the whole board is filled because a complete wall blocking the other player would be a winning path itself.2) Having an additional dot on the board can never hurt your position.3) If there existed a strategy for the second player to win, the first player could move anywhere and then on their second move start using the second player's strategy.

So, now red only needs to connect the d5 dot to the top and the d3 dot to the bottom.By symmetry, we can look at just connecting the d5 dot to the top, and the d3 dot can be connected to the bottom in the same way.

If Blue plays on 3, Red plays on 7, and then on Red's next move, either 2 or 4 will be free to connect to connecting Red to the top.If Blue plays on 7, Red plays on either 6 or 8, and then on Red's next move if Red played on 6, either 1 or 2 will be free to connect to or if red played on 8, either 4 or 5 will be free to connect to. If Blue plays on 1,2,4,5,6 or 8 Red can play at 7 and then on the next move, at least one of 2,3, or 4 will be free to connect to.If Blue doesn't play on a numbered spot, Red can just play as if Blue had played on a numbered spot.If Blue plays on the other half of the board, Red responds following this same strategy but flipped.

This game is very similar to hex https://en.wikipedia.org/wiki/Hex_(board_game). You might find it more interesting. It is still a theoretical win for the first player, but since the board is slightly less connected, the winning strategies are less obvious and immediate.